Analytic Geometry
Unit 1: Similarity,
Congruence, and Proofs
Excerpts from Georgia Department of
Education Webinar May 19, 2013
melissa.stewart@hallco.org
May 2013
Warm-Up
In the following diagram, two triangles are shaded. Based on the
information given, decide whether there is enough information to
prove that the two triangles are congruent.
The two triangles are congruent by SAS:
We have AX ≅ CX and DX ≅ BX since the diagonals of a
parallelogram bisect each other, and ∠AXD ≅ ∠CBX since they are
vertical angles.
Alternatively, the two triangles are congruent by ASA: ∠DAX ≅
∠BCX and ∠ADX ≅ ∠CBX since they are opposite interior angles.
AD ≅ BC since opposite sides of a parallelogram are congruent
melissa.stewart@hallco.org
May 2013
What’s the main idea of Unit 1?
• Understand congruence in terms of
rigid motions.
• Understand similarity in terms of
similarity transformations.
• Prove theorems involving similarity.
• Prove geometric theorems.
• Make geometric constructions.
melissa.stewart@hallco.org
May 2013
Concepts & Skills to Maintain from Previous Grades
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Understand and use reflections, translations, and rotations.
Define the following terms: circle, bisector, perpendicular and parallel.
Solve multi-step equations.
Understand angle sum and exterior angle of triangles.
Know angles created when parallel lines are cut by a transversal.
Know facts about supplementary, complementary, vertical, and
adjacent angles.
Solve problems involving scale drawings of geometric figures.
Draw geometric shapes with given conditions.
Understand that a two-dimensional figure is congruent to another if
the second can be obtained from the first by a sequence of rotations,
reflections, and translations.
Draw polygons in the coordinate plane given coordinates for the
vertices.
Websites to help with the above:
http://www.crctlessons.com/
www.aplusmath.com
www.aaamath.com
melissa.stewart@hallco.org
May 2013
Enduring Understandings from this Unit
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Given a center and a scale factor, verify experimentally, that when
dilating a figure in a coordinate plane, a segment of the pre-image
that does not pass through the center of the dilation, is parallel to its
image when the dilation is preformed. However, a segment that
passes through the center remains unchanged.
Given a center and a scale factor, verify experimentally, that when
performing dilations of a line segment, the pre-image, the segment
which becomes the image is longer or shorter based on the ratio given
by the scale factor.
Use the idea of dilation transformations to develop the definition of
similarity.
Given two figures determine whether they are similar and explain their
similarity based on the equality of corresponding angles and the
proportionality of corresponding sides.
Use the properties of similarity transformations to develop the criteria
for proving similar triangles: AA.
Use AA, SAS, SSS similarity theorems to prove triangles are similar.
Prove a line parallel to one side of a triangle divides the other two
proportionally, and its converse.
Prove the Pythagorean Theorem using triangle similarity.
Use similarity theorems to prove that two triangles are congruent.
Use descriptions of rigid motion and transformed geometric figures to
predict the effects rigid motion has on figures in the coordinate plane.
Knowing that rigid transformations preserve size and shape or distance
and angle, use this fact to connect the idea of congruency and develop
the definition of congruent.
Use the definition of congruence, based on rigid motion, to show two
triangles are congruent if and only if their corresponding sides and
corresponding angles are congruent.
Use the definition of congruence, based on rigid motion, to develop
and explain the triangle congruence criteria: ASA, SSS, and SAS.
Prove vertical angles are congruent.
Prove when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent.
Prove points on a perpendicular bisector of a line segment are exactly
those equidistant from the segment’s endpoints.
Prove the measures of interior angles of a triangle have a sum of
180º.
Prove base angles of isosceles triangles are congruent.
melissa.stewart@hallco.org
May 2013
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Prove the segment joining midpoints of two sides of a triangle is
parallel to the third side and half the length.
Prove the medians of a triangle meet at a point.
Prove properties of parallelograms including: opposite sides are
congruent, opposite angles are congruent, diagonals of a
parallelogram bisect each other, and conversely, rectangles are
parallelograms with congruent diagonals.
Copy a segment and an angle.
Bisect a segment and an angle.
Construct perpendicular lines, including the perpendicular bisector of a
line segment.
Construct a line parallel to a given line through a point not on the line.
Construct an equilateral triangle so that each vertex of the equilateral
triangle is on the circle.
Construct a square so that each vertex of the square is on the circle.
Construct a regular hexagon so that each vertex of the regular
hexagon is on the circle.
melissa.stewart@hallco.org
May 2013
Examples & Explanations
1.
Show that there is a rotation of the plane which does not move D and which
maps B’ to E.
melissa.stewart@hallco.org
May 2013
Show that there is a reflection of the plane which does not move D or E and
which maps C’’ to F.
2. The triangle in the upper left is reflected over a line to the triangle in the
lower right. Using a compass and straightedge, determine the line of
reflection.
melissa.stewart@hallco.org
May 2013
3.
melissa.stewart@hallco.org
May 2013
 The student edition for Unit 1 can be
found at
https://www.georgiastandards.org/C
ommon-Core/Pages/Math-9-12.aspx
On the left side, please look under
mathematics. Then, the right side has
a pull-down menu to access the units.
 Additional parent guides will be
posted to the parent resource page on
http://www.hallco.org/boe/index.ph
p (right hand menu) as they become
available.
melissa.stewart@hallco.org
May 2013